Spatial and observational homogeneities of the galaxy distribution in standard cosmologies

Context. Empirical verification of the geometrical concept of homogeneity of the standard relativistic cosmology considering its various definitions of distance and emphasizing that astronomical observations occur along the past light cone. Aims. To present an analysis about the physical consequences of the distinction between the usual concept of spatial homogeneity (SH), as defined by the Cosmological Principle, and the concept of observational homogeneity (OH), arguing that OH is in principle falsifiable by means of astronomical observations, whereas verifying SH is only possible indirectly. Methods. Simulated observational counts of cosmological sources (galaxies) are produced by means of a generalized number-distance expression that can be specialized to produce either the counts of the Einstein-de Sitter (EdS) cosmology, which has SH by construction, or other types of counts, which do, or do not, have OH by construction. Expressions for observational volumes are derived with the various cosmological distance definitions in the EdS cosmological model. Then the observational volumes and simulated counts are used to derive differential densities. Plots are presented showing the behavior of these densities for increasing redshift values. Results. Simulated counts that have OH by construction do not always exhibit SH features. The reverse situation is also true. In addition, simulated counts with no OH features at low redshift start showing OH characteristics at high redshift. The comoving distance seems to be the only distance definition where both SH and OH appear simultaneously, but even though with some limitations. Conclusions. The results show that observations indicating possible lack of OH do not necessarily falsify the standard Friedmannian cosmology, meaning that this cosmology will not necessarily always produce observable homogeneous densities. The general conclusion that can be draw is that the use of different cosmological distances in the characterization of the galaxy distribution lead to significant ambiguities in reaching conclusions about the behavior of the large-scale galaxy distribution in the Universe.